Question

Sa se arate ca dacă x+y=z atunci:

cos^2 x + cos^2 y + cos^2 z - 2cos x cos y cos z = 1​

Answer 1

Răspuns:

$cos^2x+cos^2y+cos^2z-2cosx*cosy*cosz=1$

$cos^2x+cos^2y+cos^2z-(cos(x+y)+cos(x-y))*cosz=1$

$cos^2x+cos^2y+cos^2z-cos^2z-cos(x-y)*cos(x+y)=1$

$cos^2x+cos^2y-\frac{cos(x-y+x+y)+cos(x-y-x-y)}{2}=1$

$\frac{2cos^2x+2cos^2y-cos(2x)-cos(2y)}{2}=1$

$\frac{2cos^2x+2cos^y-cos^2x+sin^2x-cos^2y+sin^2y}{2}=1$

$\frac{sin^2x+cos^2x+sin^2y+cos^2y}{2}=1$

$\frac{1+1}{2}=\frac{2}{2}=1$ c.c.t.d.